Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $r = \dfrac{-p^2 + 2p + 3}{p^2 - 8p + 15} \times \dfrac{p - 3}{5p + 5} $
Explanation: First factor out any common factors. $r = \dfrac{-(p^2 - 2p - 3)}{p^2 - 8p + 15} \times \dfrac{p - 3}{5(p + 1)} $ Then factor the quadratic expressions. $r = \dfrac {-(p + 1)(p - 3)} {(p - 3)(p - 5)} \times \dfrac {p - 3} {5(p + 1)} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac { -(p + 1)(p - 3) \times (p - 3)} { (p - 3)(p - 5) \times 5(p + 1)} $ $r = \dfrac {-(p + 1)(p - 3)(p - 3)} {5(p - 3)(p - 5)(p + 1)} $ Notice that $(p - 3)$ and $(p + 1)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac {-(p + 1)(p - 3)\cancel{(p - 3)}} {5\cancel{(p - 3)}(p - 5)(p + 1)} $ We are dividing by $p - 3$ , so $p - 3 \neq 0$ Therefore, $p \neq 3$ $r = \dfrac {-\cancel{(p + 1)}(p - 3)\cancel{(p - 3)}} {5\cancel{(p - 3)}(p - 5)\cancel{(p + 1)}} $ We are dividing by $p + 1$ , so $p + 1 \neq 0$ Therefore, $p \neq -1$ $r = \dfrac {-(p - 3)} {5(p - 5)} $ $ r = \dfrac{-(p - 3)}{5(p - 5)}; p \neq 3; p \neq -1 $